We study the stability of a modified (with variation in the nonlinearity parameter) “super-twisting” algorithm. The analysis is based on majorizing the trajectories of the system with an arbitrary nonlinearity parameter by the trajectories of systems of the classical “super-twisting” algorithm. Stability conditions for the modified systems are obtained, as well as estimates for the size of the stability domain depending on system parameters.
We consider a special system of integral equations of convolution type with a monotone convex nonlinearity naturally arising when searching for stationary or limit states in various dynamic models of applied nature, for example, in models of the spread of epidemics, and prove theorems stating the existence or absence of a nontrivial bounded solution with limits at (pm infty ) depending on the values of these limits and on the structure of the matrix kernel of the system. We also study the uniqueness of such a solution assuming that it exists. Specific examples of systems whose parameters satisfy the conditions stated in our theorems are given.
We consider a guaranteed control problem for a nonlinear distributed equation of diffusion type. The problem is essentially to construct a feedback control algorithm ensuring that the solution of a given equation tracks the solution of a similar equation subjected to an unknown disturbance. The case in which a discontinuous unbounded function can be a feasible disturbance is studied. We solve the problem under conditions of inaccurate measurement of solutions of each of the equations at discrete instants of time and indicate a solution algorithm robust under information noise and calculation errors.
We study a model of a predator–prey system with possible infection of prey in the form of a three-dimensional system of ordinary differential equations. Using the localization method of compact invariant sets, the existence of an attractor is proved and a compact positively invariant set is found that estimates its position. The conditions for the extinction of populations and the existence of equilibria are found. A numerical method for finding a Hopf bifurcation of the inner equilibrium is proposed and an example of an arising stable limit cycle is given.
The problem of constructing the graph of states of a switched affine system closed by a static state feedback is considered. To solve this problem, a constructive algorithm based on the study of the consistency of systems of linear algebraic inequalities is proposed.
We consider the optimal control problem of minimizing the terminal cost functional for a dynamical system whose motion is described by a differential equation with Caputo fractional derivative. The relationship between the necessary optimality condition in the form of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives is studied. It is proved that the costate variable in the Pontryagin maximum principle coincides, up to sign, with the fractional coinvariant gradient of the optimal result functional calculated along the optimal motion.
We consider autonomous differential inclusions with nonlinear boundary conditions. Sufficient conditions for the existence of solutions in the class of absolutely continuous functions are obtained for these inclusions. It is shown that the corresponding existence theorem applies to the Cauchy problem and the antiperiodic boundary value problem. The result is used to derive a new mean value inequality for continuously differentiable functions.
We propose a new approach to the solvability of ordinary as well as partial differential equations in the theory of linear differential equations and also in the theory of integral equations.
We consider a discrete-time-invariant system with multiplicative noise with implementation in the state space. The exogenous disturbance is chosen from the class of time-invariant ergodic sequences of nonzero colorness. We consider the level of mean anisotropy of the exogenous disturbance to be bounded by a known value. Conditions for the anisotropic norm to be bounded by a given number are obtained in terms of solving a matrix system of inequalities with a convex constraint of a special type. It is demonstrated how, on the basis of the obtained conditions, to construct a static state control that ensures the minimum value of the anisotropic norm of the system enclosed by this control.
A model of interaction between the human immunodeficiency virus and the human immune system is considered. Equilibria in the state space of the system and their stability are analyzed, and the ultimate bounds of the trajectories are constructed. It has been proved that the local asymptotic stability of the equilibrium corresponding to the absence of disease is equivalent to its global asymptotic stability. The loss of stability is shown to be caused by a transcritical bifurcation.
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